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Open Bosonic No Ghost
Old-covariant open-string physical quotients at D=26, intercept one, all finite oscillator levels and nonzero on-shell momentum; p=0 is excluded.
Statement
For the standard flat open bosonic string at and , every nonzero on-shell momentum p, and the physical inner product induced by pthm-st-open-oldcovariant-brst-equivalence, the old-covariant quotient has positive-definite induced inner product. The zero-momentum sector is excluded.
Statement references
David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§2.1–2.1.2 (PDF file pp.35–38; printed pp.28–31); printed pp.28–31); printed pp.28–31); printed pp.28–31); printed pp.28–31); printed pp.28–31); printed pp.28–32); old-covariant conditions and indefinite ambient form. Full positivity proof is not supplied by this locator.
Facts & Assumptions
Given: The standard critical flat open bosonic model, any fixed finite oscillator level and nonzero on-shell real momentum.
Bosonic Covariant State Space defines and its radical quotient.
Critical Dimension Flat Bosonic String supplies in the stated model.
Nonzero Momentum Pplus Chart supplies the form-preserving nonzero- chart.
Open Oldcovariant BRST Equivalence proves the actual natural isometric bijection from the old-covariant quotient onto relative ghost-number-one cohomology, with kernel exactly the radical before quotienting.
Open Bosonic BRST No Ghost proves positive definiteness of that relative cohomology with the same Hermitian form, at every finite level and nonzero on-shell momentum.
Proof
Use [F3] to work in a nonzero- chart. The actual map in [F4] identifies each quotient class with a relative BRST class and preserves its Hermitian form; the critical values are [F2]. Thus a nonzero old-covariant class maps to a nonzero BRST class.
By [F5], that nonzero target class has strictly positive norm. Isometry gives the same strictly positive norm to the original class. Spatial-rotation transport in [F3] proves the result for every admitted nonzero momentum. The radical kernel and all-level completeness are the proved map in [F4], not a separate assumed DDF completeness assertion.